Automorphic tensor products and cuspidal cohomology of the ${\mathrm{GL}}_4$

Abstract

In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\mathrm{GL}}_4({\mathbb{A}}_{\mathbb{Q}})$ to cuspidal cohomology of the ${\mathrm{GL}}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\mathrm{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\mathrm{GL}}_2$ and the automorphic tensor product of two representations of ${\mathrm{GL}}_2$ can not be equal (up to a twist by a character of ${\mathrm{GL}}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.

14 pages

References in corpus (3)

• A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)
• Algebraic cycles on Hilbert modular fourfolds and poles of L-functions
• On the growth of cuspidal cohomology of ${\mathrm{GL}}_4$